Thursday, October 11th, 2018

Time: 4:30-6:00 p.m.  Place: Jeffery Hall 422

Speaker: Pei-Lun Tseng (Queen's University)

Title: Right Hilbert A-modules

Abstract:  Last week, we gave a sketch of proving the GNS construction. We will start to introduce the matrices over a C*-algebra this week, which is an application of the GNS construction. Then, we will begin the new topic: Right Hilbert A-modules. We will deduce the process to construct the inner product on a right Hilbert A-module H. As long as we have the inner product, we can consider the orthogonality on H and compare the different between the scalar case and $A$-valued case.

Free Probability and Random Matrices Seminar Webpage:

Thursday, September 27th, 2018

Time: 4:30-6:00 p.m.  Place: Jeffery Hall 422

Speaker: Jamie Mingo (Queen's University)

Title: Additive Convolution and Subordination

Abstract:  Last week I reviewed the basic facts of scalar free independence. This week I will continue with a new convolution of probability measures called the free additive convolution. We will use a tool from complex analysis called subordination to get our convolution. It will then lead to a discussion of conditional expectation which will be the basis of operator valued freeness.

Free Probability and Random Matrices Seminar Webpage:

Wednesday, May 23rd, 2018

Time: 3:30-5:00 p.m.  Place: Jeffery Hall 319

Speaker: Benson Au (Berkeley)

Title: Rigid structures in the universal enveloping traffic space

Abstract:  For a tracial $*$-probability space $(\mathcal{A}, \varphi)$, Cébron, Dahlqvist, and Male constructed an enveloping traffic space $(\mathcal{G}(\mathcal{A}), \tau_\varphi)$ that extends the trace $\varphi$. The CDM construction provides a universal object that allows one to appeal to the traffic probability framework in generic situations, prioritizing an understanding of its structure.

We show that $(\mathcal{G}(\mathcal{A}), \tau_\varphi)$ comes equipped with a canonical free product structure, regardless of the choice of $*$-probability space $(\mathcal{A}, \varphi)$. If $(\mathcal{A}, \varphi)$ is itself a free product, then we show how this additional structure lifts into $(\mathcal{G}(\mathcal{A}), \tau_\varphi)$. Here, we find a duality between classical independence and free independence.

We apply our results to study the asymptotics of large (possibly dependent) random matrices, generalizing and providing a unifying framework for results of Bryc, Dembo, and Jiang and of Mingo and Popa. This is joint work with Camille Male.

Free Probability and Random Matrices Seminar Webpage:

Tuesday, April 3rd, 2018

Time: 3:30-5:00 p.m.  Place: Jeffery Hall 319

Speaker: Pei-Lun Tseng (Queen's University)

Title: Linearization trick of infinitesimal freeness II

Abstract:  Last week, we introduced how to find a linearization for a given selfadjoint polynomial and showed some properties of this linearization. We will continue our discussion this week and introduce the operator-valued Cauchy transform. Then, we will show the algorithm for finding the distribution of $P$ where $P$ is a selfadjoint polynomial with selfadjoint variables $X$ and $Y$. Based on this method, we will discuss how to extend this algorithm for finding infinitesimal distribution for $P$.

Free Probability and Random Matrices Seminar Webpage:

Tuesday, March 20th, 2018

Time: 3:30-5:00 p.m.  Place: Jeffery Hall 319

Speaker: Mihai Popa (University of Texas, San Antonio)

Title: Permutations of Entries and Asymptotic Free Independence for Gaussian Random Matrices

Abstract:  Since the 1980's, various classes of random matrices with independent entries were used to approximate free independent random variables. But asymptotic freeness of random matrices can occur without independence of entries: in 2012, in a joint work with James Mingo, we showed the (then) surprising result that unitarily invariant random matrices are asymptotically (second order) free from their transpose. And, in a more recent work, we showed that Wishart random matrices are asymptotically free from some of their partial transposes. The lecture will present a development concerning Gaussian random matrices. More precisely, it will describe a rather large class of permutations of entries that induces asymptotic freeness, suggesting that the results mentioned above are particular cases of a more general theory.

Free Probability and Random Matrices Seminar Webpage: